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In this paper, the stable problem for differential-algebraic systems is investigated by a convex op-timization approach. Based on the Lyapunov functional method and the delay partitioning approach, some delay and its time-derivative dependent stable criteria are obtained and formulated in the form of simple linear matrix inequalities (LMIs). The obtained criteria are dependent on the sizes of delay and its time-derivative and are less conservative than those produced by previous approaches.

Differential-algebraic systems, also referred to as singular systems, descriptor systems or generalized state-space systems, arise in a variety of practical systems such as chemical processes, nuclear reactors, biological systems, electrical networks and economy systems. Differential-algebraic systems include not only dynamic equations but also static equations [

Because of the extensive applications in many practical systems, a great number of fundamental notions and results in control and system theory based on standard state-space systems have been extended successfully to differential-algebraic systems. In recent years, much attention has been focused on stability, robust stability and

This article deals with the problem of asymptotic stability for a class of linear differential-algebraic system with time-varying delay. The obtained criteria depend not only on the upper bound but also on the lower bound of the delay derivative. Based on the Lyapunov functional method and the delay partitioning approach, some delay and its time-derivative dependent stable criteria are obtained. One numerical example is provided to demonstrate the effectiveness of the proposed results. All the developed results are in the LMI framework which makes them more interesting since the solutions are easily obtained using existing powerful tools like the LMI toolbox of Matlab or any equivalent tool.

Notation: Throughout this paper,

Consider the following differential-algebraic system:

where

_{1}, d_{2} are constants. The initial con- dition is given by

The following definition, lemmas and notation are introduced, which will be used in the proof of the main results.

Definition 1 ( [

2) System (1) is said to be impulse-free if

Lemma 1 ( [

Theorem 1 System (1) is asymptotically stable for all differentiable delays

for

Proof. The proof of this theorem is divided into two parts. First, we prove that the results, when

First of all, we divide the delay interval

where

Consider the following Lyapunov functional:

where

In addition, we define the continuous function

When

To obtain the main results, we consider the following three cases:

Case 1:

Using the fact that

where

derivative of

Letting

to (5) gives

for some scalar

where

Inequality (8) contains two variables which make it difficult to solve by LMI tool. In order to overcome this difficulty, we seek the sufficient conditions for inequality (8). When

and

where

Note that we have omitted the zero row and zero column in

Thus,

When

The LMI (11) implies (9), since

Thus,

Case 2:

By the definition of the characteristic function

and letting

for some scala

where

Similar to the case I, we obtain the results when

From Case 1 and Case 2, we have

for some scalar

Case 3:

Therefore,

Now the asymptotic stability of system (1) can not be obtained yet, since the existence and uniqueness of a solution to system (1) are not always guaranteed and the system may have undesired impulsive behavior. In the following, we will prove that the above-mentioned results ensure the regular and impulse-free. It follows form (8), that

Pre- and post-multiplying (15) by

Since

Similar to (17), we define

Then, using

where

When the matrix E is nonsingular, the result in this case can be obtained by setting E equal to I with the appropriate transformations. The corresponding result is given by the following corollary:

Corollary 1 System (1) with

and

When

Corollary 2 System (1) is asymptotically stable for all differentiable delays

and

Remark 1. It should be pointed out that if

results. In addition, the better results may be obtained if we divide the delay interval

segments.

In this section, a numerical example will be presented to show the validity of the main results derived above.

Example Consider the following linear differential-algebraic system described by systems (1) with

For

In this paper, the asymptotic stability of differential-algebraic system with time-varying delay has been investi- gated. Some delay and its time-derivative dependent asymptotically stable criteria have been obtained by de- composing time-varying delay in a convex set. The obtained criteria depend not only on the upper but also on the lower bound of the delay derivative. One numerical example has been given to illustrate the effectiveness of the proposed main results.

We thank the Editor and the referee for their comments. This work was supported by A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department (16ZA0146) and the Doctoral Research Foundation of Southwest University of Science and Technology (13zx7141).

Hui Liu,Yucai Ding, (2016) Delay and Its Time-Derivative Dependent Stable Criterion for Differential-Algebraic Systems. Applied Mathematics,07,1124-1133. doi: 10.4236/am.2016.710100